Integral representation for functionals
defined on SBDp in dimension two
September 28, 2015
Sergio Conti1 , Matteo Focardi2 , and Flaviana Iurlano1
1
Institut für Angewandte Mathematik, Universität Bonn
53115 Bonn, Germany
2
DiMaI, Università di Firenze
50134 Firenze, Italy
We prove an integral representation result for functionals
with growth conditions which give coercivity on the space
SBDp (Ω), for Ω ⊂ R2 . The space SBDp of functions whose
distributional strain is the sum of an Lp part and a bounded
measure supported on a set of finite H1 -dimensional measure
appears naturally in the study of fracture and damage models.
Our result is based on the construction of a local approximation by W 1,p functions. We also obtain a generalization of
Korn’s inequality in the SBDp setting.
1
Introduction
The direct methods of Γ-convergence are of paramount importance in studying variational limits and relaxation problems since their introduction in the
seminal paper by Dal Maso and Modica [27]. They focus on the study of abstract limiting functionals F (u, A), obtained for instance using Γ-convergence
arguments; one key ingredient is the proof of an integral representation for
F (u, A). Here u : Ω → RN is an element of a suitable function space X (Ω),
and A runs in the class A(Ω) of open subsets of a given open set Ω ⊂ Rn . The
notion of variational functional is at the heart of the matter: F , regarded as
depending on the couple (u, A) ∈ X (Ω) × A(Ω), has to satisfy suitable lower
semicontinuity, locality and measure theoretic properties (for more details
see properties (i)-(iii) in Theorem 1.1). The specific growth conditions of the
functional determine the natural functional space in which the function u
lies. Under these assumptions F (u, A) can be written as an integral over the
domain of integration A with respect to a suitable measure. The integrands
may depend on x, u(x) and ∇u(x), and possibly on other local quantities
of u, such as higher order or distributional derivatives. Furthermore, as first
shown in some cases in [28] and then generalized in [10], the corresponding
int_repr10.tex
1
[September 28, 2015]
energy densities can be characterized in terms of cell formulas, i.e. asymptotic
Dirichlet problems on small cubes or balls involving F itself, with boundary
data depending on the local properties of u.
Integral representation results have been obtained in several contexts with
increasing generality: starting with the pioneering contribution by De Giorgi
for limits of area-type integrals [29], it has been extended to functionals
defined first on Sobolev spaces in [38, 15, 12, 14, 13] and on the space of
functions with Bounded Variation in [24, 8], and then to energies defined on
partitions in [2] and on the subspace SBV in [11] (we refer to [13, 25, 10, 9]
for a more exhaustive list of references). The global method for relaxation
introduced and developed in [10, 9] provides a general approach that unifies
and extends the quoted results.
We address the integral representation of functionals defined on the subspace SBDp (Ω) of the space BD(Ω) in two dimensions. The space of functions of bounded deformation BD(Ω) is characterized by the fact that the
symmetric part of the distributional gradient Eu := (Du + DuT )/2 of u ∈
L1 (Ω, Rn ) is a bounded Radon measure, namely
BD(Ω) := {u ∈ L1 (Ω; Rn ) : Eu ∈ M(Ω; Rn×n
sym )},
where Ω ⊆ Rn is an open set, see [39, 40, 4]. BD and its subspaces SBD
and SBDp constitute the natural setting for the study of plasticity, damage
and fracture models in a geometrically linear framework [39, 40, 41, 6, 35].
In particular, SBDp is the set of BD functions such that the strain Eu can
be written as the sum of a function in Lp (Ω, Rn×n ) and a part concentrated
on a rectifiable set with finite Hn−1 -measure, see [7, 16, 17, 19].
For functionals with linear growth defined on SBD an integral representation result was obtained by Ebobisse and Toader [31]. These functionals,
however, lack coercivity on the relevant space. The situation of functionals
defined on SBDp and with corresponding growth properties is open. We give
here a solution in two dimensions.
Theorem 1.1. Let Ω ⊂ R2 be a bounded Lipschitz set, p ∈ (1, ∞), F :
SBDp (Ω) × B(Ω) → [0, ∞) be such that
(i) F (u, ·) is a Borel measure for any u ∈ SBDp (Ω);
(ii) F (·, A) is lower semicontinuous with respect to the strong L1 (Ω, R2 )convergence for any open set A ⊂ Ω;
(iii) F (·, A) is local for any open set A ⊂ Ω;
int_repr10.tex
2
[September 28, 2015]
(iv) There are α, β > 0 such that for any u ∈ SBDp (Ω), any B ∈ B(Ω),
ˆ
ˆ
p
α( |e(u)| dx +
(1 + |[u]|)dH1 ) ≤ F (u, B)
Ju ∩B
ˆB
ˆ
p
≤β( (|e(u)| + 1)dx +
(1 + |[u]|)dH1 ).
(1.1)
Ju ∩B
B
Then there are two Borel functions f : Ω × R2 × R2×2 → [0, ∞) and g :
Ω × R2 × R2 × S 1 → [0, ∞) such that
ˆ
ˆ
F (u, B) =
f (x, u(x), ∇u(x))dx +
g(x, u− (x), u+ (x), νu (x))dH1 .
B
B∩Ju
(1.2)
Above and throughout the paper we will refer to the book [5] and to
the papers [4, 7] for the notation and results about BV and BD spaces,
respectively. In particular, B(Ω) is the family of Borel subsets of Ω.
The proof of Theorem 1.1, which is given in Section 4, follows the general
strategy introduced in [10, 9]. Their approach was based on a Poincaré-type
inequality in SBV by De Giorgi, Carriero and Leaci, which is not known
in SBDp (see [30, 5]). Our main new ingredient is the construction of an
approximation by W 1,p functions, discussed in Section 3, which permits to
bypass the De Giorgi-Carriero-Leaci inequality. The approximation is done so
that the function is only modified outside a countable set of balls with small
area and perimeter. In each ball, we give a construction of a W 1,p extension
for the SBDp function by constructing a finite-element approximation on a
countable mesh, which is chosen depending on the function u, see Section 2.
Our W 1,p approximation result also leads naturally to the proof of the
following variant of Korn’s inequality for SBDp functions.
Theorem 1.2. Let Ω ⊂ R2 be a connected, bounded, Lipschitz set and let
p ∈ (1, ∞). Then there exists a constant c, depending on p and Ω, with the
following property: for every u ∈ SBDp (Ω) there exist a set ω ⊂ Ω of finite
perimeter, with H1 (∂ω) ≤ cH1 (Ju ), and an affine function a(x) = Ax + b,
with A ∈ R2×2 skew-symmetric and b ∈ R2 , such that
ku − akLp (Ω\ω,R2 ) ≤ cke(u)kLp (Ω,R2×2 ) ,
k∇u − AkLp (Ω\ω,R2×2 ) ≤ cke(u)kLp (Ω,R2×2 ) .
This improves a result of [33] to the sharp exponent. Variants of the first
inequality were first obtained in [18, 32].
The construction of Section 3 turns out to be crucial also in proving existence for the Griffith fracture model, generalizing [30] to the case of linearized
elasticity in dimension two. This will be the object of the forthcoming paper
[20].
int_repr10.tex
3
[September 28, 2015]
2
Approximation of SBDp functions with small
jump set
In this Section we prove the following approximation result.
Theorem 2.1. Let n = 2, p ∈ [1, ∞). There exists η > 0 and c̃ > 0 such
that if J ∈ B(B2r ), for some r > 0, satisfies
H1 (J) < 2rη,
(2.1)
then there exist R ∈ (r, 2r) for which the following holds: for every u ∈
SBDp (B2r ) with H1 (Ju ∩ B2r \ J) = 0 there exists φ(u) ∈ SBDp (B2r ) ∩
W 1,p (BR , R2 ) such that
(i) H1 (Ju ∩ ∂BR ) = 0;
ˆ
ˆ
q
|e(φ(u))| dx ≤ c̃
(ii)
|e(u)|q dx, for every q ∈ [1, p];
BR
BR
(iii) ku − φ(u)kL1 (BR ,R2 ) ≤ c̃R|Eu|(BR );
(iv) u = φ(u) on B2r \ BR , H1 (Jφ(u) ∩ ∂BR ) = 0;
√
(v) if u ∈ L∞ (B2r , R2 ), then kφ(u)kL∞ (B2r ,R2 ) ≤ 2 kukL∞ (B2r ,R2 ) .
Proof. We follow an idea of [23, 22].
Arguing as in [23, Lemma 4.3] we first claim that there exists R ∈ (r, 2r)
such that for δk := R 2−k we have
H1 (J ∩ ∂BR ) = 0,
H1 (J ∩ (BR \ BR−δk )) < 20ηδk , for every k ∈ N.
(2.2)
(2.3)
To prove this, we first observe that (2.2) holds for almost every R, therefore
it suffices to show that (2.3) holds on a set of positive measure. We consider
the family of intervals
{[R − δk , R] : H1 (J ∩ (BR \ BR−δk )) ≥ 20ηδk }
and we define I as the union of all intervals of the family, with R ∈ (r, 2r),
k ∈ N. By Vitali’s covering theorem, there exists a countable set (Ri , ki )i∈N
such that the corresponding intervals [Ri − δki , Ri ] are pairwise disjoint and
cover at least one fifth of I. Therefore by (2.1) we obtain
X
X
2rη > H1 (J ∩ B2r ) ≥
H1 (J ∩ (BRi \ BRi −δki )) ≥
20ηδki .
i∈N
i∈N
int_repr10.tex
4
[September 28, 2015]
Rk+1
Rk
Rk−1
Figure 1: Sketch of the construction of the grid in the proof of Theorem 2.1.
Since δki = L1 ([Ri − δki , Ri ]), we conclude that L1 (I) < r. This proves the
existence of R such that (2.2) and (2.3) hold, which is fixed for the rest of
the proof.
, sin 2πj
), j = 1, . . . , 2k . We
We define Rk := R − δk and xk,j := Rk (cos 2πj
2k
2k
0
say that xk,j and xk0 ,j 0 are neighbors if either k = k and j = j 0 ± 1, working
modulo 2k , or (up to a permutation) k = k 0 + 1 and j ∈ {2j 0 − 1, 2j 0 , 2j 0 + 1},
again modulo 2k . This gives a decomposition of BR into countably many
triangles, whose angles are uniformly bounded away from 0 and π, see Figure
1.
We shall construct φ(u) as a linear interpolation on a triangulation whose
vertices are slight modifications of xk,j . Following the idea of [22, Proposition
2.2], we next show how to construct the modified triangulation. We start
off considering two neighboring points x and y in {xk,j }k,j , connected by the
segment Sx,y ⊂ B Rk+1 \BRk−1 for some k, and notice that c1 δk ≤ |x−y| ≤ c2 δk
for some c1 ∈ (0, 1), c2 > 1 independent from k. Let α := c1 /(8c2 ) and
consider the convex envelope
Ox,y := conv (B(x, αδk ) ∪ B(y, αδk )).
(2.4)
Let ax,y denote the infinitesimal rigid movement appearing in the Poincaré’s
inequality for u on the set
Qx,y := {ξ ∈ BR : dist(ξ, Sx,y ) < |x − y|/(8c2 )}.
Given ϑ ∈ (0, 1), let us prove that for η sufficiently small and c̃ sufficiently
large, depending only on ϑ, there exists a subset F ⊂ B(x, αδk )×B(y, αδk )
2 ×L2 )(F )
with (L
< ϑ, such that for every (x, y) ∈
/ F the one-dimensional
L2 (Bαδk )2
ν
section uz has the following properties:
int_repr10.tex
5
[September 28, 2015]
z + Rν ⊥
x
y
S x,
x
y
z
y
ν
Figure 2: Slice along the line of direction ν = (x − y)/|x − y| passing through
z in the proof of Theorem 2.1.
(P1) uνz ∈ SBV (sx,y );
(P2) H0 (Juνz ) = 0, so that uνz ∈ W 1,1 (sx,y );
ˆ
ˆ
c̃
ν 0
|(uz ) |dt ≤
(P3)
|e(u)|dx0 ;
δk Ox,y
sx,y
(P4) |u(ξ) − ax,y (ξ)| ≤
c̃
|Eu|(Qx,y ), for ξ = x, y.
δk
Here uνz (t) := u(z + tν) · ν is the slice of u along the line of direction
ν :=
x−y
,
|x − y|
(2.5)
and passing through
z := (Id − ν ⊗ ν)x ∈ Rν ⊥ ∩ (x + Rν)
(2.6)
where Rν ⊥ is the linear space orthogonal to ν, and sx,y ⊂ R is the segment
such that z + sx,y ν = Sx,y , see Figure 2.
In order to obtain property (P1) we first define the measure µν,z := |uνz |L1
and we observe that
ˆ
µν,z (sx,y )dx dy
(2.7)
B(x,αδk )×B(y,αδk )
ˆ
ˆ
2
≤ cδk
dx
µν,z ((Ox,y )νz )dH1 (ν),
{|ν−ν|≤c(α)}
B(x,αδk )
int_repr10.tex
6
[September 28, 2015]
by the change of variables y = x + sν, where ν is defined as ν with x, y in
place of x, y and (Ox,y )νz ⊂ R corresponds to Ox,y ∩ (z + Rν). By Fubini’s
theorem the last term in the previous inequality is less than or equal to
ˆ
ˆ
1
3
dH (ν)
µν,z ((Ox,y )νz )dH1 (z) ≤ cδk3 µ(Ox,y ),
(2.8)
cδk
{|ν−ν|≤c(α)}
ν⊥
´
where µ(Ox,y ) := Ox,y |u|dx0 . By (2.7) and (2.8) the set F1 ⊂ B(x, αδk )×B(y, αδk )
of points (x, y), for which the inequality
µν,z (sx,y ) >
2
c̃
µ(Ox,y )
δk
2
)(F1 )
holds, satisfies (LL2×L
< ϑ/16, for c̃ large enough, depending on ϑ and α.
(Bαδk )2
For (x, y) ∈ B(x, αδk )×B(y, αδk ) \ F1 we now repeat the argument in
(2.7) and (2.8) above redefining
µν,z := |D(uνz )|.
We find that for (x, y) out of a small (in the previous sense) set F10 one has
uνz ∈ SBV (sx,y ) and
|D(uνz )|(sx,y ) ≤
c̃
|Eu|(Ox,y ).
δk
As for property (P2), we use (2.7) and (2.8) with µν,z := H0 (Juνz ∩ sx,y )
and µ := H1 (Ju ∩ Ox,y ). Now (2.1) implies
ˆ
H0 (Juνz ∩ sx,y )dx dy ≤ cδk4 η,
B(x,αδk )×B(y,αδk )
and hence the set F2 of points (x, y) for which H0 (Juνz ∩ sx,y ) > 1/2 is also
small in the previous sense, for η small enough. Note that this is the only
step which requires the hypothesis on the dimension n = 2.
Analogously properties (P3) and (P4) can be derived. From the argument
above it is straightforward that for many points x ∈ B(x, αδk ), still in the
sense of a large ϑ-fraction of B(x, αδk ), there are many points y ∈ B(y, αδk )
for which (x, y) ∈
/ F.
Let us construct now the modified grid with an iterative process (see
also [22, Proposition 3.4]). We will use the notation Bi to indicate the balls
B(xk,j , αδk ), lexicographically ordered.
We start by fixing a point x0 ∈ B0 for which there are many good choices
in each neighboring ball. We next select x1 ∈ B1 among the points which are
int_repr10.tex
7
[September 28, 2015]
good choices for x0 and which have many good choices in each neighboring
subsequent ball Bi , i ≥ 2. Iterating the process, the point xm ∈ Bm will
be taken among the good choices for the neighboring previously fixed xi ,
i < m, and with the property that have many good choices in the neighboring
subsequent Bi , i > m. Since each ball can have at most seven neighbors, at
each step we select xm avoiding just a small subset of Bm .
We call S the set of points obtained by this process and we construct
a new triangulation, with x, y neighbors if and only if x̄, ȳ are neighbors.
Notice that again
c1 δk ≤ |x − y| ≤ c2 δk ,
(2.9)
for every couple of neighboring points x, y, with the same k as for the corresponding reference points x and y, and suitable c1 , c2 > 0 independent from
k. We finally define φ(u) as the linear interpolation between the values of
u(x), x ∈ S on each triangle of the triangulation.
Fixed a triangle T and any couple of its vertices x, y, we compute a
component of the constant matrix e(φ(u)) on T by
ˆ
(φ(u)(x) − φ(u)(y)) · ν
e(φ(u))ν · ν =
(2.10)
= − (uνz )0 dt,
|x − y|
sx,y
where ν and z are defined in (2.5) and (2.6). We used the fact that u and
φ(u) agree on x and y and that u is W 1,1 (sx,y ) by the choice of x and y. By
(2.10), (2.9), and property (P3) above it follows
ˆ
c̃
|e(φ(u))ν · ν| ≤ 2
|e(u)|dx0 ,
δk C
where C is defined in (2.4). We recall that here and henceforth c̃ can possibly
change. Letting ν vary among the directions of the sides of T , we obtain a
control on the whole |e(φ(u))| thanks to (2.9)
ˆ
c̃
|e(φ(u))| ≤ 2
|e(u)|dx0 ,
(2.11)
δk C T
where CT denotes the convex envelope
CT := conv (∪B(x, αδk ))
and the union is taken over the three vertices x in the old triangulation
corresponding to the three vertices of T . We remark that B(x, αδk ) ⊂ BRk+1 \
BRk−1 for all x ∈ ∂BRk , therefore the sets CT have finite overlap.
int_repr10.tex
8
[September 28, 2015]
We are ready to prove property (ii). By Jensen’s inequality and (2.11)
we have for 1 ≤ q ≤ p
ˆ
ˆ
q
q
0
2
q
2
|e(φ(u))| dx = L (T )|e(φ(u))| ≤ c̃L (CT ) − |e(u)|dx0
T
CT
ˆ
≤ c̃
|e(u)|q dx0 ,
CT
and finally summing up on all triangles T we get the conclusion.
In order to prove properties (iii) and (iv) we estimate
ˆ
ˆ
ˆ
0
0
|u − φ(u)|dx ≤
|u − ax,y |dx + |ax,y − φ(u)|dx0 ,
T
(2.12)
T
T
where T is again a triangle of the modified triangulation with vertices x, y, z,
x, y, z denote the three corresponding vertices of the old triangulation, ax,y
is the infinitesimal rigid motion appearing in the Poincaré’s inequality for u
on Qx,y (see item (P4) above).
Let us study first the second term in (2.12). Since ax,y − φ(u) is affine, it
achieves its maximum on a vertex ξ of T , therefore
ˆ
|ax,y − φ(u)|dx0 ≤ cδk2 |ax,y (ξ) − φ(u)(ξ)| = cδk2 |ax,y (ξ) − u(ξ)|.
T
Notice that if ξ = z then
cδk2 |ax,y (ξ) − u(ξ)| ≤ cδk2 |ax,y (ξ) − ax,ξ (ξ)| + cδk2 |ax,ξ (ξ) − u(ξ)|
ˆ
≤ c
|ax,y − ax,ξ |dx0 + cδk |Eu|(Qx,ξ ),(2.13)
B(x,αδk )
where we used the fact that ax,ξ , ax,y are affine and item (P4) above; if ξ ∈
{x, y} then only the second term appears.
By (2.12)-(2.13), the triangular inequality, and Poincaré’s inequality we
conclude
ˆ
|u − φ(u)|dx0 ≤ cδk |Eu|(QT ),
(2.14)
T
where QT := Qx,y ∪ Qy,z ∪ Qz,x . Finally summing up over T we obtain
property (iii).
We prove now property (iv), property (v) holding true by construction.
We define φ(u) := u outside B R and know that φ(u) ∈ W 1,p (BR , R2 ) ∩
SBD(B2r ). It remains to prove that the traces on ∂BR coincide, or, equivalently, that H1 (Jφ(u) ∩ BR ) = 0. Let ψk ∈ C ∞ (BR ) be such that ψk = 0
on BRk , ψk = 1 in a neighborhood of ∂BR , and |∇ψk | ≤ c/δk . We define
int_repr10.tex
9
[September 28, 2015]
vk := (u − φ(u))ψk ∈ SBD(BR ) and we prove that vk → 0 strongly in BD,
this implying in turn vk |∂BR → 0 in L1 (∂BR , R2 ) in the sense of traces and
therefore property (iv). Clearly
ˆ
ˆ
|vk |dx ≤
|u − φ(u)|dx → 0
BR
BR \BRk
by the dominated convergence theorem. Finally, using (2.14) and the fact
that the triangles have finite overlap,
ˆ
c
|u − φ(u)|dx
|Evk |(BR ) ≤ |E(u − φ(u))|(BR \ BRk ) +
δk BR \BR
k
≤ c̃|E(u − φ(u))|(BR \ BRk ),
the last term tends to 0 and this concludes the proof of property (iv).
3
Regularity of SBDp functions with small
jump set
We first discuss how SBDp functions can be approximated by W 1,p functions
locally away from the jump set (Section 3.1), and then how they can be
approximated by piecewise W 1,p functions around the jump set (Section 3.3).
Our approximation result also leads to the Korn inequality stated in Theorem
1.2. The key ingredient for all these results is the construction of Theorem
2.1. Throughout the section η ∈ (0, 1) will be the constant from Theorem 2.1
and n = 2.
3.1
Approximation of SBDp functions with W 1,p functions
We shall use that the construction of Theorem 2.1, using a suitable covering
argument, permits to approximate SBDp functions by W 1,p functions which
coincide away from a small neighborhood of the jump set. The neighborhood
is the union of countably many balls, such that each of them contains an
amount of jump set proportional to the radius. Before discussing the covering
argument in Proposition 3.2, we show that (away from the boundary) almost
any point of the jump set is the center of a ball with the appropriate density.
int_repr10.tex
10
[September 28, 2015]
Lemma 3.1. Let s ∈ (0, 1). Let J ∈ B(B2ρ ), for some ρ > 0, be such
that H1 (J) < η(1 − s)ρ then for H1 -a.e. x ∈ J ∩ B2sρ there exists a radius
rx ∈ (0, (1 − s)ρ) such that
H1 J ∩ ∂Brx (x) = 0,
(3.1)
(3.2)
η rx ≤ H1 J ∩ Brx (x) ≤ H1 J ∩ B2rx (x) < 2 η rx .
Proof. We fix x ∈ J∩B2sρ , choose λx ∈ (ρ, 2ρ) such that H1 J∩∂Bλx/2k (x) =
0 for all k ∈ N, and define
rx := max{λx/2k : k ∈ N, H1 J ∩ Bλx/2k (x) ≥ η λx 2−k }.
The set is nonempty for H1 -almost every x because η < 1. The estimates
(3.2) hold by definition. To conclude that rx < (1 − s)ρ it is enough to notice
that the opposite inequality would give the ensuing contradiction
H1 (J) ≥ H1 J ∩ Brx (x) ≥ η rx ≥ (1 − s)η ρ > H1 (J).
We are now ready to prove the main result of the section via a covering
argument, Lemma 3.1, and Theorem 2.1.
Proposition 3.2. Let p ∈ (1, ∞), n = 2. There exists a universal constant
c > 0 such that if u ∈ SBDp (B2ρ ), ρ > 0, satisfies
H1 (Ju ∩ B2ρ ) < η (1 − s)ρ
for η ∈ (0, 1) as in Theorem 2.1 and some s ∈ (0, 1), then there is a countable
family F = {B} of closed balls of radius rB < (1 − s)ρ, each contained in
B(1+s)ρ , and a field w ∈ SBDp (B2ρ ) such that
P
P
(i) ρ−1 F L2 B + F H1 ∂B ≤ c/η H1 (Ju ∩ B2ρ );
(ii) H1 Ju ∩ ∪F ∂B = H1 (Ju ∩ B2sρ ) \ ∪F B = 0;
(iii) w = u L2 -a.e. on B2ρ \ ∪F B;
(iv) w ∈ W 1,p (B2sρ , R2 ) and H1 (Jw \ Ju ) = 0;
(v)
ˆ
ˆ
p
|e(u)|p dx;
|e(w)| dx ≤ c
∪F B
and there exists a skew-symmetric matrix A such that
ˆ
ˆ
p
|∇u − A| dx ≤ c(p)
|e(u)|p dx;
B2sρ \∪F B
int_repr10.tex
(3.3)
∪F B
(3.4)
B2ρ
11
[September 28, 2015]
(vi) ku − wkL1 (B,R2 ) ≤ c rB |Eu|(B), for every B ∈ F;
(vii) if, additionally, u ∈ L∞ (B2ρ , R2 ) then w ∈ L∞ (B2ρ , R2 ) with
kwkL∞ (B2ρ ,R2 ) ≤ ckukL∞ (B2ρ ,R2 ) .
Proof. By Lemma 3.1 we find a family F 0 of open balls covering H1 -a.e.
Ju ∩ B2sρ that satisfies (3.1) and (3.2). Setting J = Ju , to every B ∈ F 0
we associate a new ball B ∗ ⊂ B with the properties (i)-(v) of Theorem 2.1.
Let F ∗ be the family of the new balls B ∗ , this is still a cover of J. Further,
the balls B ∗ can be taken to be closed. By the Besicovitch covering theorem
[5, Theorem 2.17] there are ξ countable subfamilies Fj0 = {Bji }i∈N of disjoint
balls. Therefore, setting F := ∪ξj=1 Fj0 we have H1 (Ju ∩ B2sρ ) \ ∪F B = 0.
In addition, by (3.1) the first condition in item (ii) is satisfied as well, so that
(ii) is established. Furthermore,
(3.2)
2π X 1
H Ju ∩ B
η B∈F
B∈F
B∈F
2π
2π
≤ξ H1 Ju ∩ ∪B∈F B ≤ ξ H1 (Ju ∩ B2ρ ).
η
η
P 2
P
The volume estimate follows since rB ≤ ρ implies
rP
rB . We
B ≤ ρ P
2
remark that a quadratic volume estimate also follows by
rB ≤ ( rB )2 .
Let φ(u) be the function given by Theorem 2.1 on the balls of the first
family F10 and define for every h ∈ N a function
(
φ(u) B1i , i ≤ h
w1h :=
u
otherwise
X
H1 (∂B) =2π
X
rB ≤
such that w1h ∈ SBDp (B2ρ ), w1h ∈ W 1,p (∪i≤h B1i ; R2 ) with w1h = u L2 -a.e. on
B2ρ \ ∪i≤h B1i and H1 (Jw1h \ Ju ) = 0. In addition by item (ii) in Theorem 2.1
ˆ
ˆ
|e(w1h )|p
ˆ
p
∪i≤h B1i
ˆ
ˆ
p
≤ c̃
|e(u)| dx +
∪i≤h B1i
|e(u)|p dx
ˆ
p
|e(u)| dx ≤ (1 + c̃)
|e(u)|p dx, (3.5)
|e(φ(u))| dx +
dx =
B2ρ
B2ρ \∪i≤h B1i
B2ρ \∪i≤h B1i
B2ρ
ˆ
and
|Ew1h |(B2ρ )
int_repr10.tex
≤ |Eu| B2ρ \
∪i≤h B1i
12
|e(u)| dx.
+ c̃
∪i≤h B1i
[September 28, 2015]
Moreover, recalling that the B1i ’s are disjoint and that w1h−1 = u on B1h , item
(iii) in Theorem 2.1 gives
kw1h − w1h−1 kL1 (B2ρ ;R2 ) = kw1h − ukL1 (B1h ;R2 ) ≤ c ρ |Eu|(B1h ),
in turn implying that for all h ≥ k ≥ 1
kw1h
−
w1k kL1 (B2ρ ;R2 )
≤
h
X
kw1i − w1i−1 kL1 (B1h ;R2 ) ≤ c ρ |Eu| ∪k+1≤i≤h B1i .
i=k+1
Thus, w1h → w1 in L1 (B2ρ ; R2 ) with
(
φ(u) ∪F10 B
w1 :=
u
otherwise.
The BD compactness theorem then yields that w1 ∈ BD(B2ρ ). In turn, by
(3.5) and since H1 (Jw1h \ Ju ) = 0, the SBD compactness theorem implies
that actually w1 ∈ SBDp (B2ρ ) (see also [26, Theorem 11.3]). Furthermore,
since
H1 Jw1h ∩ ∪F10 B = H1 (Ju ∩ ∪i≥h+1 B1i ,
we may conclude that
H1 Jw1 ∩ ∪F10 B ≤ lim inf H1 Jw1h ∩ ∪F10 B = 0,
h
and therefore w1 ∈ W 1,p (∪F10 B, R2 ). Finally, by construction w1 = u L2 -a.e.
on B2ρ \ ∪F10 B and H1 (Jw1 \ Ju ) = 0.
By iterating the latter construction, for all 1 < k ≤ ξ and for every h ∈ N
we find
(
φ(wk−1 ) Bki , i ≤ h
wkh :=
wk−1
otherwise
such that wkh ∈ SBDp (B2ρ ), wkh ∈ W 1,p (∪i≤h Bki ; R2 ), wkh = wk−1 L2 -a.e. on
B2ρ \ ∪i≤h Bki , H1 (Jwkh \ Jwk−1 ) = 0. In addition, arguing as above, wkh → wk
in L1 (B2ρ , R2 ) with
(
φ(wk−1 ) ∪Fk0 B
wk :=
wk−1
otherwise,
wk ∈ SBDp (B2ρ ), wk ∈ W 1,p (∪j≤k ∪Fj0 B; R2 ), wk = wk−1 L2 -a.e. on B2ρ \
∪Fk0 B and H1 (Jwk \ Jwk−1 ) = 0.
int_repr10.tex
13
[September 28, 2015]
Set w := wξ , then w ∈ SBDp (B2ρ ), w ∈ W 1,p (∪F B; R2 ), w = u L2 a.e. on B2ρ \ ∪F B, H1 (Jw \ Ju ) = 0. Iterating estimate (3.5), inequality
(3.3) follows at once with c := max{1 + c̃, 2π} ξ, with c̃ the constant in
Theorem 2.1. Korn’s inequality (3.4) follows now immediately by (iii), (iv),
and (3.3).
Finally, it is clear that also items (vi) and (vii) are satisfied in view of
properties (iii) and (v) in Theorem 2.1.
3.2
Korn’s inequality in SBDp
Proof of Theorem 1.2. By standard scaling and covering arguments it suffices
to prove the assertion for a special Lipschitz domain. Precisely, let ϕ : R → R
Lipschitz with min ϕ[(−1, 1)] = 2, and set U := {x : x1 ∈ (−2, 2), x2 ∈
(−2, ϕ(x1 ))}, and U int := {x : x1 ∈ (−1, 1), x2 ∈ (−1, ϕ(x1 ))}. It suffices
to show that for any u ∈ SBDp (U ) there are ω with H1 (∂ω) + |ω|1/2 ≤
cH1 (Ju ) and an affine function a : R2 → R2 such that ku − akLp (U int \ω,R2 ) +
k∇u − ∇akLp (U int \ω,R2 ) ≤ cL,p ke(u)kLp (U,R2×2 ) , with c depending on p and the
Lipschitz constant L of ϕ. Obviously we can assume H1 (Ju ) to be small.
Let qj := xj + (−rj /2, rj /2)2 and Qj := xj + (−rj , rj )2 , and assume
that u ∈ SBDp (Qj ) √
obeys H1 (Ju ∩ Qj ) ≤ ηrj /8. By Proposition 3.2 with
ρ := rj /2 and s := 1/ 2 and Poincaré’s inequality there are ωj and aj affine
with rj H1 (∂ωj )+|ωj |1/2 ≤ cH1 (Ju ∩Qj ) and rj−1 kuj −aj kLp (qj \ωj ,R2 ) +k∇uj −
∇aj kLp (qj \ωj ,R2×2 ) ≤ cp ke(u)kLp (Qj ,R2×2 ) , with a constant which depends only
on the exponent p.
To pass to the estimate on U int one uses a Whitney cover with pairs of
open cubes qj and Qj such that the exterior ones have finite overlap and
the interior ones form a cover, as done for example in proving the nonlinear
Korn’s inequality in [34, Theorem 3.1]. Following [33], if H1 (Ju ∩Qj ) ≥ ηrj /8
we define Pj := xj + (−rj , rj ) × (−rj , ∞) ∩ U , otherwise Pj = ∅ and ωj , aj are
obtained as above. Notice that H1 (Pj ) ≤ cL rj by the properties of Lipschitz
functions and of the Whitney covering.
Then it suffices to apply the weighted Poincaré inequality, as done in [34,
Theorem 3.1] and [33, Theorem 4.2]. By the properties of the covering, for
neighboring pairs of squares |qj ∩ qi | ≥ cri2 , and if η is not too large this
gives a bound on the difference of the two affine functions. One then defines
a∗ ∈ C ∞ (U, R2 ) using a partition of unit subordinated to the cover {qj }, and
obtains
ˆ
(ϕ(x1 ) − x2 )p |D2 a∗ |p (x)dx ≤ cL,p ke(u)kpLp (U,R2 ) .
U int \∪j Pj
Since the cube Q0 = (−2, 2)2 was not removed one has a∗ = a0 in q0 and
int_repr10.tex
14
[September 28, 2015]
application of the one-dimensional weighted Poincaré inequality to a∗ (x1 , ·)
leads to the assertion, with ω := ∪j (Pj ∪ ωj ) and a := a0 . Equivalently, in
the last step one may use a Poincaré or Korn inequality on John domains,
as done in [33, Theorem 4.2].
We remark that the nonoptimality of the exponent in [33, Theorem 4.2]
is only consequence of the nonoptimal local estimate employed there (see [33,
Theorem 3.1]).
3.3
Reflection
In this subsection we establish a technical result instrumental for the identification of the surface energy density in Section 4.3. To this aim, given
u ∈ SBDp (Ω) and a point x0 ∈ Ju we set
(
u+ (x0 ) if hx − x0 , νx0 i > 0,
ux0 (x) :=
(3.6)
u− (x0 ) if hx − x0 , νx0 i < 0.
Lemma 3.3. Let p ∈ (1, ∞), u ∈ SBDp (Ω), Ω ⊂ R2 open. For H1 -a.e.
x0 ∈ Ju and any ρ > 0 sufficiently small there is vρ ∈ SBDp (B2ρ (x0 )) ∩
SBV p (Bρ (x0 ), R2 ) such that:
1
(i) lim H1 (Jvρ \ Ju ∩ Bρ (x0 )) = 0;
ρ→0 ρ
ˆ
1
(ii) lim
|∇vρ |p dx = 0;
ρ→0 ρ B (x )
ρ 0
1 2
L ({x ∈ Bρ (x0 ) : u 6= vρ }) = 0;
ρ2
ˆ
1
(iv) lim 2
|vρ − u|dx = 0;
ρ→0 ρ
Bρ (x0 )
ˆ
1
(v) lim p+1
|vρ − ux0 |p dx = 0;
ρ→0 ρ
Bρ (x0 )
ˆ
1
(vi) lim
|[vρ ] − [u]|dH1 = 0.
ρ→0 ρ B (x )∩J
ρ 0
vρ
(iii) lim
ρ→0
int_repr10.tex
15
[September 28, 2015]
1
Proof. Since Ju is (H1 , 1) rectifiable, there exists a sequence (Γi )∞
i=1 of C
1
∞
1
curves such that H (Ju \ ∪i=1 Γi ) = 0. For H -a.e. x0 ∈ Ju we have
ˆ
1
lim
(|[u]| + 1)dH1 = |[u](x0 )| + 1,
ρ→0 2ρ J ∩B (x )
ˆ u ρ 0
1
lim
(|[u]| + 1)dH1 = |[u](x0 )| + 1,
ρ→0 2ρ J ∩Γ∩B (x )
u
ρ 0
for one of the aforementioned curves Γ. Therefore
ˆ
1
lim
(|[u]| + 1)dH1 = 0
ρ→0 2ρ (J 4Γ)∩B (x )
u
ρ 0
(3.7)
and for ρ small Γ separates B6ρ (x0 ) into two connected components. It is not
restrictive to assume that Γ ∩ B6ρ (x0 ) is the graph of a function h ∈ C 1 (R).
Moreover the following properties hold H1 -a.e. x0 ∈ Ju
ˆ
1
|e(u)|p dx = 0,
(3.8)
lim
ρ→0 ρ B (x )
ρ 0
1
lim |Eu|(Bρ (x0 )) = |[u] νu |(x0 ),
(3.9)
ρ→0 2ρ
ˆ
1
|u − u± (x0 )|dx = 0.
(3.10)
lim
ρ→0 ρ2 B (x )∩{±(x −h(x ))>0}
ρ 0
2
1
For simplicity we next assume that the point x0 = 0 satisfies all the
previous properties (3.7)-(3.10), with h(0) = h0 (0) = 0. We also set τρ :=
khkL∞ (B6ρ ) and note that τρ /ρ → 0 as ρ → 0. We now define the reflections
of u with respect to the lines {x2 = ±τρ }, in the sense of [37, Lemma 1].
More precisely, define ũ+
ρ on the set B2ρ ∩ {x2 < τρ } by
(
(ũ+
ρ )1 (x1 , x2 ) := −2u1 (x1 , 3τρ − 2x2 ) + 3u1 (x1 , 2τρ − x2 )
(ũ+
ρ )2 (x1 , x2 ) := 4u2 (x1 , 3τρ − 2x2 ) − 3u2 (x1 , 2τρ − x2 )
p
and by u otherwise in B2ρ . Note that ũ+
ρ ∈ SBD (B2ρ ) and that
1 1
H (Jũ+ρ ∩ B2ρ ) = 0,
ρ→0 2ρ
ke(ũ+
ρ )kLp (B2ρ ,R2×2 ) ≤ cke(u)kLp (B6ρ ,R2×2 ) ,
lim
(3.11)
(3.12)
for a universal constant c. Using a similar reflection we define ũ−
ρ in B2ρ ∩
−
{(x1 , x2 ) : x2 > −τρ } and we set ũρ := u otherwise in B2ρ .
By (3.7) and (3.11) for ρ small we have that ũ±
ρ satisfy the hypotheses of
Proposition 3.2 on B2ρ with s = 1/2. Thus, there exist wρ± ∈ SBDp (B2ρ ) ∩
int_repr10.tex
16
[September 28, 2015]
W 1,p (Bρ , R2 ), for which properties (i)-(vii) hold true. Finally let us define
vρ ∈ SBDp (B2ρ ) by
(
wρ+ in B2ρ ∩ {x2 > h(x1 )},
vρ :=
wρ− in B2ρ ∩ {x2 < h(x1 )}.
Since wρ± ∈ W 1,p (Bρ , R2 ) we obtain vρ ∈ SBV p (Bρ , R2 ) with
Dvρ Bρ =∇wρ+ L2 Bρ ∩ {x2 > h(x1 )} + (wρ+ − wρ− ) ⊗ νΓ H1 Γ ∩ Bρ
+ ∇wρ− L2 Bρ ∩ {x2 < h(x1 )}.
We next check that vρ satisfies the properties in the statement in the ball
Bρ . Property (i) comes straightforwardly from (3.7) and from the fact that
Jvρ ⊂ Γ. Moreover (3.12), (3.3), and (3.8) yield
ˆ
1
lim
|e(wρ± )|p dx = 0.
(3.13)
ρ→0 ρ B
ρ
As for property (iii), we observe that
1 2
τρ c
L ({Bρ : u 6= vρ }) ≤ lim (c + H1 ((Ju \ Γ) ∩ B6ρ )) = 0,
2
ρ→0 ρ
ρ→0
ρ
ρ
lim
where we have used Proposition 3.2 (i) and (3.7).
Let us now prove property (iv). By the definition of vρ and ũ±
ρ and by
triangular inequality we obtain
1
ρ2
ˆ
Bρ
ˆ
1
−
+ 2
|ũ+
− u|dx+
ρ Bρ ∩{h(x1 )<x2 <τρ } ρ
Bρ ∩{h(x1 )<x2 }
ˆ
1
−
−
|wρ − ũρ |dx + 2
|ũ−
ρ − u|dx. (3.14)
ρ
Bρ ∩{x2 <h(x1 )}
Bρ ∩{−τρ <x2 <h(x1 )}
1
ρ2
ˆ
1
ρ2
|vρ − u|dx ≤
ˆ
|wρ+
ũ+
ρ |dx
By the definition of wρ+ and Proposition 3.2 (vi) we can estimate
ˆ
1
c
c
+
|wρ+ − ũ+
ρ |dx ≤ |E ũρ |(B2ρ ) ≤ |Eu|(B6ρ \ Γ).
2
ρ Bρ
ρ
ρ
By (3.7) and (3.8) we conclude that the first term of (3.14) tends to 0.
Clearly, the same argument can be applied to the third term there. So, it
int_repr10.tex
17
[September 28, 2015]
remains to treat the second term in (3.14), being the fourth one similar. By
triangular inequality and a change of variable we infer
ˆ
1
|ũ+ − u|dx ≤
ρ2 Bρ ∩{h(x1 )<x2 <τρ } ρ
ˆ
ˆ
1
1
+
+
|ũ − u (x0 )|dx + 2
|u+ (x0 ) − u|dx ≤
ρ2 Bρ ρ
ρ Bρ ∩{h(x1 )<x2 }
ˆ
c
|u+ (x0 ) − u|dx,
ρ2 B6ρ ∩{h(x1 )<x2 }
and the last term tends to 0 by (3.10), hence property (iv) follows.
Let us prove now property (v). By Korn’s inequality and Poincaré’s
inequality in W 1,p , there exists an affine function aρ (x) := dρ + βρ x such that
ˆ
ˆ
c
1
+
p
|e(wρ+ )|p dx.
|w − aρ | dx ≤
(3.15)
ρp+1 Bρ ρ
ρ Bρ
We first claim that
lim dρ = u+ (x0 ).
(3.16)
ρ→0
Let ωρ+ := Bρ ∩ {u = wρ+ } ∩ {x2 > h(x1 )}. Since |ωρ+ |/ρ2 → π/2, and aρ is
affine, by [21, Lemma 4.3] we obtain, for ρ small,
ˆ
ˆ
c
c
+
+
kaρ − u (x0 )kL∞ (Bρ+ ,R2 ) ≤ 2
|w − aρ |dx + 2
|u − u+ (x0 )|dx.
ρ ωρ+ ρ
ρ ωρ+
The right hand side above is infinitesimal by (3.15), (3.13) and (3.10), thus
we conclude
lim sup |dρ − u+ (x0 )| ≤ lim kaρ − u+ (x0 )kL∞ (Bρ+ ,R2 ) = 0,
ρ→0
ρ→0
which proves (3.16).
Next we prove that
lim ρ|βρ |p = 0,
(3.17)
ρ→0
lim ρ
ρ→0
1−p
p
|dρ − u+ (x0 )| = 0.
(3.18)
To establish (3.17), we fix δ > 0 small and we consider ρ̂ such that
1 ˆ
p1
|e(wρ+ )|p dx < δ,
for ρ ≤ ρ̂,
ρ Bρ
int_repr10.tex
18
(3.19)
[September 28, 2015]
note that this is possible by (3.13). For ρ < ρ̂ we define ρk := (2k ρ) ∧ ρ̂ and
we adopt the notation k in place of ρk for the subscriptions. As above, using
[21, Lemma 4.3] and the triangular inequality we infer
kak − ak+1 kL∞ (Bρ+ ,R2 ) ≤
k
ˆ
ˆ
p−1
c
c
p
+
+
|w
−
a
|dx
≤
cδρ
|w
−
a
|dx
+
k+1
k
k+1
k ,
+
ρk 2 {u=wk+ } k
ρ2k+1 {u=wk+1
}
where the last estimate follows by Hölder’s inequality, (3.15), and (3.19).
Therefore
p−1
|dk − dk+1 | ≤ kak − ak+1 kL∞ (Bρ+ ,R2 ) ≤ cδρk p ,
(3.20)
k
and hence once more by [21, Lemma 4.3] and by the triangular inequality we
conclude
−1
|βk − βk+1 | ≤ cδρk p .
Collecting these estimates as k varies we obtain
k̂−1
p
X
|βk − βk+1 | ≤ cδ p + cρ|β̂|p ,
ρ|βρ |p ≤ ρ |β̂| +
k=0
where k̂ is the first index such that ρk̂ = ρ̂ and β̂ := βk̂ = βρ̂ . This proves
(3.17) as ρ → 0 and δ → 0.
We next prove (3.18). Similarly to the previous estimate, summing (3.20)
gives
|dρ − dρ̂ | ≤ cδ ρ̂(p−1)/p
for all 0 < ρ < ρ̂ ≤ ρδ , with δ arbitrary and ρδ depending only on δ. Taking
ρ → 0 and using (3.16) yields
ρ̂(1−p)/p |u+ (x0 ) − dρ̂ | ≤ cδ
which, since δ was arbitrary, proves (3.18) and therefore (v).
At this point we turn to property (ii). Korn’s inequality implies that
2
k∇wρ+ kLp (Bρ ,R2×2 ) ≤ k∇wρ+ − βρ kLp (Bρ ,R2×2 ) + c ρ /p |βρ |
2
≤ c ke(wρ+ )kLp (Bρ ,R2×2 ) + c ρ /p |βρ |,
where c > 0 is a universal constant. This, together with (3.13) and (3.17)
and the corresponding estimates for wρ− , implies property (ii).
int_repr10.tex
19
[September 28, 2015]
We finally show property (vi). Note that by the trace theorem we have
ˆ
1
|v ± − u± |dH1 ≤
ρ Γ∩Bρ ρ
ˆ
c
c
|vρ − u|dx + |E(vρ − u)|(Bρ \ Γ) ≤
2
ρ Bρ
ρ
ˆ
ˆ
ˆ
ˆ
c
c
c
c
|vρ − u|dx +
|e(vρ )|dx +
|e(u)|dx +
|[u]|dH1
ρ2 Bρ
ρ Bρ
ρ Bρ
ρ Ju \Γ
and all terms in the last expression approach 0 respectively by (iv), (3.8),
(3.13) and (3.7).
4
4.1
Integral representation
Preliminaries
In this Section we prove Theorem 1.1, along the lines of [9, Section 2.2].
Before starting we specify that property (ii) means that if uj , u ∈ SBDp (Ω)
obey uj → u in L1 (Ω, R2 ), then F (u, A) ≤ lim inf j→∞ F (uj , A) for any open
set A. Property (iii) means that if u, v ∈ SBDp (Ω) obey u = v L2 -a.e. in
A, then F (u, A) = F (v, A). The functions f and g are defined in (4.1) and
(4.2) below.
The family of balls contained in Ω is denoted by
A∗ (Ω) := {Bε (x) : x ∈ Ω, ε > 0} .
Let B ∈ A∗ (Ω). We can identify any u ∈ SBDp (B) with its zero extension
uχB ∈ SBDp (Ω), and correspondingly write F (u, B) for F (uχB , B). By
locality, for any other extension the value of the functional is the same.
For B ∈ A∗ (Ω) we define
m(u, B) := inf{F (w, B) : w ∈ SBDp (B), w = u around ∂B}
where the condition w = u around ∂B means that a ball B 0 ⊂⊂ B exists, so
that w = u on B \ B 0 . For δ > 0, A ∈ A(Ω), we set
δ
m (u, A) := inf{
∞
X
m(u, Bi ) : Bi ∈ A∗ , Bi ∩ Bj = ∅, Bi ⊂ A,
i=1
diam (Bi ) < δ, µ(A \
∞
[
Bi ) = 0} ,
i=1
int_repr10.tex
20
[September 28, 2015]
where µ := L2 Ω + (1 + |[u]|)H1 (Ju ∩ Ω).
Since δ 7→ mδ (u, A) is decreasing, we can define
m∗ (u, A) := lim mδ (u, A).
δ→0
Moreover, we set
f (x0 , u0 , ξ) := lim sup
ε→0
m(u0 + ξ(· − x0 ), Bε (x0 ))
L2 (Bε )
(4.1)
m(ux0 ,a,b,ν , Bε (x0 ))
,
2ε
(4.2)
g(x0 , a, b, ν) := lim sup
ε→0
where ux0 ,a,b,ν is defined as
(
a if hx − x0 , νi > 0,
ux0 ,a,b,ν (x) :=
b if hx − x0 , νi < 0.
Lemma 4.1. For all u ∈ SBDp (Ω) and A ∈ A(Ω), F (u, A) = m∗ (u, A).
Proof. By definition, m(u, B) ≤ F (u, B) for any ball B. Since F (u, ·) is a
measure, we obtain mδ (u, A) ≤ F (u, A) for any δ > 0. Therefore m∗ (u, A) ≤
F (u, A).
To prove the converse inequality, let δ > 0, pick countably many balls Biδ
as in the definition of mδ (u, A), such that
∞
X
m(u, Biδ ) < mδ (u, A) + δ .
i=1
By the definition of m there are functions viδ ∈ SBDp (Biδ ) such that viδ = u
around ∂Biδ and F (viδ , Biδ ) ≤ m(u, Biδ ) + δL2 (Biδ ). We define
∞
X
v δ :=
viδ χBiδ + uχN0δ
i=1
where N0δ := Ω \ ∪i Biδ .
By the BD compactness theorem v δ ∈ BD(Ω) and by the SBD closure
theorem (see also [26, Theorem 11.3]) we conclude that v δ ∈ SBDp (Ω) and
δ
Ev =
∞
X
Eviδ
Biδ + Eu N0δ ,
i=1
with
|Ev δ | N δ = 0,
int_repr10.tex
µ(N δ ) = 0 ,
21
F (v δ , N δ ) = 0
[September 28, 2015]
where N δ := A ∩ N0δ . Further,
δ
F (v , A) =
∞
X
F (viδ , Biδ ) + F (v δ , N δ ) ≤ mδ (u, A) + δ + δL2 (A) .
i=1
We claim that v δ → u in L1 (Ω, R2 ). Since F (·, A) is lower semicontinuous,
this will imply
F (u, A) ≤ lim inf F (v δ , A) ≤ lim inf mδ (u, A) = m∗ (u, A) .
δ→0
δ→0
To prove v δ → u, we observe that by Poincaré’s inequality, diam Biδ ≤ δ, and
v δ = u on ∂Biδ we obtain
kv δ − ukL1 (Biδ ,R2 ) ≤ cδ|Ev δ − Eu|(Biδ ) .
Therefore
kv δ − ukL1 (Ω,R2 ) ≤
X
kv δ − ukL1 (Biδ ,R2 ) ≤ cδ(|Ev δ |(A) + |Eu|(A))
i
≤cδ(F (v δ , A) + F (u, A)) .
Since F (v δ , A) has a finite limit as δ → 0, this proves v δ → u in L1 (Ω, R2 ).
Lemma 4.2. For any ball Br (x0 ) ⊂ Ω and δ > 0 sufficiently small we have
(i) lim m(u, Br−δ (x0 )) = m(u, Br (x0 ));
δ→0
ˆ
(ii) m(u, Br+δ (x0 ))) ≤ m(u, Br (x0 )) + β
ˆ
β
(1 + |[u]|)dH1 .
(1 + |e(u)|p )dx +
Br+δ (x0 )\Br (x0 )
Ju ∩Br+δ (x0 )\Br (x0 )
Proof. We drop x0 from the notation. Choose vδ ∈ SBDp (Br−δ ) with vδ = u
around ∂Br−δ and F (vδ , Br−δ ) ≤ m(u, Br−δ ) + δ. We define
(
vδ (x) if x ∈ Br−δ ,
wδ (x) :=
u(x)
if x ∈ Ω \ Br−δ .
We have
m(u, Br ) ≤F (wδ , Br ) ≤ F (vδ , Br−δ ) + F (wδ , Br \ Br−δ )
≤m(u, Br−δ ) + δ
ˆ
ˆ
p
+β
(|e(u)| + 1)dx + β
(1 + |[u]|)dHn−1 .
Br \Br−δ
int_repr10.tex
Ju ∩Br \Br−δ
22
[September 28, 2015]
Since (1 + |e(u)|p )L2 + (1 + |[u]|)H1 Ju is a bounded measure, we conclude
that
m(u, Br ) ≤ lim inf m(u, Br−δ ) .
δ→0
Conversely, for any ε > 0 there is vε ∈ SBDp (Br ) with vε = u around ∂Br
and F (vε , Br ) ≤ m(u, Br ) + ε. For δ > 0 sufficiently small one has vε = u on
Br \ Br−2δ and therefore m(u, Br−δ ) ≤ F (vε , Br−δ ) ≤ m(u, Br ) + ε. Taking
first δ → 0 and then ε → 0 concludes the proof of (i). The proof of (ii) is
analogous.
Lemma 4.3. For µ-a.e. x0 ∈ Ω,
F (u, Bε (x0 ))
m(u, Bε (x0 ))
= lim
.
ε→0 µ(Bε (x0 ))
ε→0 µ(Bε (x0 ))
lim
Proof. From m(u, Bε (x0 )) ≤ F (u, Bε (x0 )) one immediately obtains
lim sup
ε→0
m(u, Bε (x0 ))
F (u, Bε (x0 ))
≤ lim sup
µ(Bε (x0 ))
µ(Bε (x0 ))
ε→0
for any x0 ∈ Ω. To prove the converse inequality, we define for t > 0 the set
Et := {x ∈ Ω : there is εh → 0 such that
F (u, Bεh (x)) > m(u, Bεh (x)) + tµ(Bεh (x)) for all h} .
From this definition one immediately has
lim inf
ε→0
F (u, Bε (x0 ))
m(u, Bε (x0 ))
≤ lim inf
+t
ε→0
µ(Bε (x0 ))
µ(Bε (x0 ))
for all x0 ∈ Ω \ Et .
If we can prove that
µ(Et ) = 0 for all t > 0
(4.3)
(u,Bε (x0 ))
then, recalling that limε→0 Fµ(B
exists µ-almost everywhere, the proof
ε (x0 ))
is concluded.
It remains to prove (4.3) for an arbitrary t > 0. For δ > 0 we define
X δ := {Bε (x) : ε < δ, B ε (x) ⊂ Ω, µ(∂Bε (x)) = 0,
F (u, Bε (x)) > m(u, Bε (x)) + tµ(Bε (x))}
and
U ∗ :=
\
{x : ∃ε > 0 s.t. Bε (x) ∈ X δ } .
δ>0
int_repr10.tex
23
[September 28, 2015]
We first show that Et ⊂ U ∗ . Let x ∈ Et . Then for any δ > 0 there is ε ∈ (0, δ)
such that F (u, Bε (x)) > m(u, Bε (x))+tµ(Bε (x)). By Lemma 4.2 the function
ε → m(u, Bε (x)) is left-continuous; F (u, Bε (x)) is left-continuous because
F (u, ·) is a measure, therefore the same inequality holds for all ε0 ∈ (ε00 , ε).
In particular, there is one which additionally obeys µ(∂Bε (x)) = 0.
It remains to showSthat µ(U ∗ ) = 0. We fix a compact set K ⊂ U ∗ and
0 < δ < η. Let U η := {Bε (x) : Bε (x) ∈ X η } and
Y δ := {Bε (x) : ε < δ, Bε (x) ⊂ U η \ K, µ(∂Bε (x)) = 0} .
By definition, X δ is a fine cover of K and Y δ of U η \ K. Therefore there are
countably many pairwise disjoint balls Bi ∈ X δ and B̂j ∈ Y δ and a set N
with µ(N ) = 0 such that
!
!
[
[
Uη =
Bi ∪
B̂j ∪ N .
i∈N
j∈N
Then
F (u, U η ) =
X
≥
X
=
X
F (u, Bi ) +
i
X
F (u, B̂j ) + F (u, N )
j
(m(u, Bi ) + tµ(Bi )) +
i
X
m(u, B̂j )
j
m(u, Bi ) +
i
X
m(u, B̂j ) + tµ(∪i Bi )
j
≥mδ (u, U η ) + tµ(K)
where in the last step we used the definition of mδ . For δ → 0, the definition
of m∗ and Lemma 4.1 give
F (u, U η ) ≥ m∗ (u, U η ) + tµ(K) = F (u, U η ) + tµ(K) .
Therefore µ(K) = 0, and by the regularity of µ we conclude µ(U ∗ ) = 0.
4.2
Bounds on the volume term
In this subsection we identify the volume energy density in the integral representation for F to be the function f defined in (4.1). Throughout the whole
subsection we consider a fixed map u ∈ SBDp (Ω). Our first result shows that
int_repr10.tex
24
[September 28, 2015]
the local volume energy density can be computed with a W 1,p -approximation
to the blow-ups of u (see (4.7–4.8) below), in the sense that
m wε , Bε (x0 )
dF (u, ·)
.
(4.4)
(x0 ) = lim
ε→0
dL2
L2 (Bε )
We shall however not need (4.4), but only the apparently more complex
version in (4.5)-(4.6). Taking a diagonal subsequence they imply (4.4).
Lemma 4.4. For L2 -almost any x0 ∈ Ω, any ε > 0, and any s ∈ (0, 1) there
are functions wεs ∈ W 1,p (Bsε (x0 ); R2 ) which obey
m wεs , Bsε (x0 )
dF (u, ·)
(4.5)
(x0 ) ≤ lim inf lim inf
s→1
ε→0
dL2
L2 (Bsε )
and
m wεs , Bs2 ε (x0 )
dF (u, ·)
lim sup lim sup
≤
(x0 )
2
L (Bsε )
dL2
s→1
ε→0
(4.6)
and which approximate the affine function y 7→ ∇u(x0 )(y − x0 ) + u(x0 ) in
the sense that
ˆ
1
lim
|e(wεs ) − e(u)(x0 )|p dx = 0
(4.7)
ε→0 ε2 B (x )
ε 0
and
lim
ε→0
1
ˆ
|wεs (x) − u(x0 ) − ∇u(x0 )(x − x0 )|p dx = 0 .
ε2+p
(4.8)
Bε (x0 )
We remark that the ball in (4.6) has radius s2 ε instead of sε. The estimate
would also hold on Bsε , the variant we chose is more convenient in the proof
of Lemma 4.7.
Proof. Let x0 ∈ Ω be such that
ˆ
1
|e(u)(x) − e(u)(x0 )|p dx = 0 ,
lim
ε→0 ε2 B (x )
ε 0
ˆ
1
lim
(1 + |[u]|)dH1 = 0 ,
ε→0 ε2 B (x )∩J
ε 0
u
and
1
lim 3
ε→0 ε
(4.9)
(4.10)
ˆ
|u(x) − u(x0 ) − ∇u(x0 )(x − x0 )|dx = 0 .
(4.11)
Bε (x0 )
int_repr10.tex
25
[September 28, 2015]
By [4, Th. 7.4], L2 -almost every x0 obeys (4.11), the other two are standard.
By (4.10), for sufficiently small ε one has H1 (Ju ∩ Bε (x0 )) ≤ η(1 − s)ε/2,
where η is the constant from Theorem 2.1. By Proposition 3.2 applied to
u − u(x0 ) − ∇u(x0 )(· − x0 ) there is w̃εs ∈ SBDp (Bε (x0 )) ∩ W 1,p (Bsε (x0 ); R2 )
with properties (i)-(vii) and we set wεs := w̃εs + u(x0 ) + ∇u(x0 )(· − x0 ). In
particular, (4.7) follows from (3.3) and (4.9), while (4.8) follows from Lemma
4.5 below applied to w̃εs , estimating the right-hand side with (4.7), (vi), and
(4.9)-(4.11).
We first prove (4.6). By the definition of m and the fact that F (wεs , ·) is
a measure follows
m(wεs , Bsε (x0 )) ≤ F (wεs , Bsε (x0 )) ≤ F (wεs , Bε (x0 )) .
Let (Bi )i∈N be the balls from Proposition 3.2. For M ∈ N we define
wεs,M := u + χ∪M
(wεs − u) .
i=1 B i
Then wεs,M ∈ SBDp (Bε (x0 )) and wεs,M → wεs in L1 as M → ∞. Further,
F (wεs,M , Bε (x0 ))
≤F (wεs,M , Bε (x0 )
\
∪M
i=1 B i )
+
≤F (u, Bε (x0 ) \ ∪M
i=1 B i ) + β
M
X
F (wεs,M , B i )
i=1
M
Xˆ
i=1
(1 + |e(wεs )|p )dx
Bi
since wεs,M = wεs is a W 1,p function on each B i . By monotonicity and lower
semicontinuity of F we obtain
F (wεs , Bε (x0 ))
≤F (u, Bε (x0 )) + β
∞ ˆ
X
i=1
2
(1 + |e(wεs )|p )dx
Bi
≤F (u, Bε (x0 )) + cL (∪i Bi )(1 + |e(u)|p (x0 ))
ˆ
+c
(|e(wεs ) − e(u)(x0 )|p )dx
Bε (x0 )
and, recalling Proposition 3.2 (i), conclude the proof of (4.6) by (4.7) and
(4.9).
It remains to prove (4.5). Let vε ∈ SBDp (Bs2 ε (x0 )) be such that vε = wεs
around ∂Bs2 ε (x0 ) and F (vε , Bs2 ε ) ≤ m(wεs , Bs2 ε (x0 )) + ε3 . We define
(
vε (x)
if x ∈ Bs2 ε (x0 )
ṽε (x) :=
s
wε (x) if x ∈ Bε (x0 ) \ Bs2 ε (x0 ) .
int_repr10.tex
26
[September 28, 2015]
By definition of m and additivity of F we obtain
m(u, Bε (x0 )) ≤F (ṽε , Bε (x0 )) = F (ṽε , Bs2 ε (x0 )) + F (ṽε , Bε (x0 ) \ Bs2 ε (x0 ))
where by locality of F and definition of vε
F (ṽε , Bs2 ε (x0 )) = F (vε , Bs2 ε (x0 )) ≤ m(wεs , Bs2 ε (x0 )) + ε3
and, since ṽε = wεs outside Bs2 ε (x0 ) and H1 (Jṽε ∩ ∂Bs2 ε (x0 )) = 0, recalling
(3.3) we obtain
ˆ
F (ṽε , Bε (x0 ) \ Bs2 ε (x0 )) ≤β
(1 + |e(wεs )|p )dx
Bε (x0 )\Bs2 ε (x0 )
ˆ
+β
(1 + |[u]|)dH1
Ju ∩Bε (x0 )\Bs2 ε (x0 )
2
≤cβL (Bε )(1 − s4 )(1 + |e(u)|p (x0 ))
ˆ
+ cβ
|e(wεs )(x) − e(u)(x0 )|p dx
B (x )
ˆ ε 0
+β
(1 + |[u]|)dH1 .
Ju ∩Bε (x0 )
Dividing by L2 (Bε ) and taking the limit ε → 0 gives
m(u, Bε (x0 ))
m(wεs , Bs2 ε (x0 ))
≤
lim
inf
+ cβ(1 − s4 )(1 + |e(u)|p (x0 )) ,
2
2
ε→0
ε→0
L (Bε )
L (Bε )
lim
where we used (4.7) and (4.10). Recalling Lemma 4.3 we obtain
dF (u, ·)
m(u, Bε (x0 ))
m(wεs , Bs2 ε (x0 ))
(x
)
=
lim
≤
lim
inf
lim
inf
.
0
ε→0
s→1
ε→0
dL2
L2 (Bε )
L2 (Bsε )
This concludes the proof of (4.5).
The next Lemma is a reverse-Hölder estimate for functions with small
0
strain, of the form kvkp ≤ rke(v)kp + kvk1 r−n/p .
Lemma 4.5. For any p ≥ 1 there is c > 0 (depending on n and p) such that
for any v ∈ W 1,p (Br ; Rn ) one has
p
ˆ
ˆ
ˆ
1
1
1
p
p
|v| dx ≤ c n
|e(v)| dx + c n+1
|v|dx .
rn+p Br
r Br
r
Br
int_repr10.tex
27
[September 28, 2015]
Proof. By scaling it suffices to consider r = 1. By Korn’s inequality there is
an affine function a such that
ˆ
ˆ
p
|v − a| dx ≤ c
|e(v)|p dx .
B1
Since a is affine,
ˆ
ˆ
p
|a| dx ≤ c
B1
B1
p
ˆ
|a|dx ≤ c
B1
ˆ
p
|v|dx
|v − a|p dx .
+c
B1
B1
A triangular inequality concludes the proof.
Lemma 4.6. For L2 -a.e. x0 ∈ Ω,
dF (u, ·)
(x0 ) ≤ f (x0 , u(x0 ), ∇u(x0 ))
dL2
where f was defined in (4.1).
Proof. Let x0 , wεs be as in Lemma 4.4, for s ∈ (0, 1). We choose vεs ∈
SBDp (Bs2 ε (x0 )) such that vεs (x) = u(x0 ) + ∇u(x0 )(x − x0 ) around ∂Bs2 ε (x0 )
and F (vεs , Bs2 ε (x0 )) ≤ m(u(x0 ) + ∇u(x0 )(· − x0 ), Bs2 ε (x0 )) + ε3 . We extend it
to R2 setting it equal to u(x0 ) + ∇u(x0 )(· − x0 ) outside Bs2 ε (x0 ) and choose
ϕ ∈ Cc∞ (Bsε (x0 )) with ϕ = 1 on Bs2 ε (x0 ) and kDϕk∞ ≤ c/(s(1 − s)ε). We
define
zεs := ϕvεs + (1 − ϕ)wεs .
We remark that zεs = vεs on Bs2 ε (x0 ) and zεs ∈ W 1,p (Bsε (x0 ) \ Bs2 ε (x0 ); R2 ).
Then
m(wεs , Bsε (x0 )) ≤F (zεs , Bsε (x0 )) ≤ F (vεs , Bs2 ε (x0 )) + F (zεs , Bsε (x0 ) \ Bs2 ε (x0 ))
≤m(u(x0 ) + ∇u(x0 )(· − x0 ), Bs2 ε (x0 )) + ε3
ˆ
(1 + |e(zεs )|p )dx .
+β
Bsε (x0 )\Bs2 ε (x0 )
In order to estimate the error term, we observe that in Bsε (x0 ) \ Bs2 ε (x0 ) one
has
∇zεs − ∇u(x0 ) = (u(x0 ) + ∇u(x0 )(· − x0 ) − wεs )∇ϕ + (1 − ϕ)(∇u(x0 ) − ∇wεs )
which implies
ˆ
(1 + |e(zεs )|p )dx ≤c(1 − s)L2 (Bsε )(1 + |e(u)|p (x0 ))
Bsε (x0 )\Bs2 ε (x0 )
ˆ
+c
|e(u)(x0 ) − e(wεs )|p dx
B (x )
ˆ sε 0
|u(x0 ) + ∇u(x0 )(· − x0 ) − wεs |p
+c
dx.
εp sp (1 − s)p
Bsε (x0 )
int_repr10.tex
28
[September 28, 2015]
Therefore
lim sup
ε→0
F (zεs , Bsε (x0 ) \ Bs2 ε (x0 ))
≤ c(1 − s)(1 + |e(u)|p (x0 ))
L2 (Bsε )
and
m(wεs , Bsε (x0 ))
m(u(x0 ) + ∇u(x0 )(· − x0 ), Bs2 ε (x0 ))
lim sup
≤ lim sup
2
L (Bsε )
L2 (Bsε )
ε→0
ε→0
+ c(1 − s)(1 + |e(u)|p (x0 ))
=s2 f (x0 , u0 , ∇u(x0 )) + c(1 − s)(1 + |e(u)|p (x0 )) .
Since s was arbitrary, this concludes the proof.
Lemma 4.7. For L2 -a.e. x0 ∈ Ω,
f (x0 , u(x0 ), ∇u(x0 )) ≤
dF (u, ·)
(x0 )
dL2
where f was defined in (4.1).
Proof. We choose x0 and wεs as in Lemma 4.4, for s ∈ (0, 1). We let vεs ∈
SBDp (Bs2 ε (x0 )) be such that vεs = wεs around ∂Bs2 ε (x0 ) and F (vεs , Bs2 ε (x0 )) ≤
m(wεs , Bs2 ε (x0 )) + ε3 , and extend it to Bsε (x0 ) setting it equal to wεs outside Bs2 ε (x0 ). We choose ϕ ∈ Cc∞ (Bsε (x0 )) with ϕ = 1 on Bs2 ε (x0 ) and
kDϕk∞ ≤ c/(s(1 − s)ε) and define
zεs := ϕvεs + (1 − ϕ)(u(x0 ) + ∇u(x0 )(x − x0 )) .
Then
m(u(x0 ) + ∇u(x0 )(· − x0 ),Bsε (x0 )) ≤ F (zεs , Bsε (x0 ))
=F (vεs , Bs2 ε (x0 )) + F (zεs , Bsε (x0 ) \ Bs2 ε (x0 ))
≤m(wεs , Bs2 ε (x0 )) + ε3 + F (zεs , Bsε (x0 ) \ Bs2 ε (x0 )) .
In order to estimate the error term, we observe that in Bsε (x0 ) \ Bs2 ε (x0 ) one
has
∇zεs − ∇u(x0 ) = −(u(x0 ) + ∇u(x0 )(· − x0 ) − wεs )∇ϕ + ϕ(∇wεs − ∇u(x0 ))
which leads as in the proof of Lemma 4.6 to
lim sup
ε→0
F (zεs , Bsε (x0 ) \ Bs2 ε (x0 ))
≤ c(1 − s)(1 + |e(u)|p (x0 )) .
L2 (Bsε )
int_repr10.tex
29
[September 28, 2015]
We conclude that for any s ∈ (0, 1)
m(u(x0 ) + ∇u(x0 )(· − x0 ), Bsε (x0 ))
L2 (Bsε )
ε→0
m(wεs , Bs2 ε (x0 ))
+ c(1 − s)(1 + |e(u)|p (x0 )) .
≤ lim sup
L2 (Bsε )
ε→0
lim sup
Since s was arbitrary, this concludes the proof.
4.3
Bounds on the surface term
In the current subsection we identify the function g in (4.2) to be the surface
energy density in the integral representation of F . As above, we work with
a fixed map u ∈ SBDp (Ω).
We first prove a technical result.
Lemma 4.8. For H1 -a.e. x0 ∈ Ju the functions v2ε ∈ SBV p (B2ε (x0 ), R2 )
introduced in Lemma 3.3 satisfy for all t ∈ (0, 2)
m(v2ε , Btε (x0 ))
dF (u, ·)
(x0 ) = lim
.
1
ε→0
dH Ju
2tε
(4.12)
Proof. It suffices to consider points x0 such that the conclusions of LemdF (u,·)
mata 3.3 and 4.3 hold true, the Radon-Nikodym derivative dH
(x0 ) exists
1
Ju
finite,
µ(Bε (x0 ))
= 1 + |[u](x0 )|,
(4.13)
lim
ε→0
2ε
and
ˆ
1 ˆ
1
p
lim
|e(u)| dx + 2
|u(x) − ux0 |dx = 0,
(4.14)
ε→0 ε B (x )
ε Bε (x0 )
ε 0
where ux0 is the piecewise constant function defined in (3.6). In view of all
these choices and thanks to Lemma 4.3 we may conclude that
F (u, Bε (x0 ))
m(u, Bε (x0 ))
dF (u, ·)
(x0 ) = lim
= lim
.
1
ε→0
ε→0
dH Ju
2ε
2ε
(4.15)
For ε > 0 small enough the function v2ε introduced in Lemma 3.3 belongs
to SBDp (B4ε (x0 )) ∩ SBV p (B2ε (x0 ), R2 ) and it satisfies properties (i)-(vi).
We set wε := v2ε , we are left with proving that for all t ∈ (0, 2)
dF (u, ·)
m(wε , Btε (x0 ))
(x0 ) ≥ lim sup
,
1
dH Ju
2tε
ε→0
int_repr10.tex
30
(4.16)
[September 28, 2015]
dF (u, ·)
m(wε , Btε (x0 ))
(x0 ) ≤ lim inf
.
1
ε→0
dH Ju
2tε
(4.17)
For the sake of notational simplicity we shall prove inequalities (4.16) and
(4.17) only for t = 1.
We start off with (4.16). Let (εj )j be a sequence such that
m(wεj , Bεj (x0 ))
m(wε , Bε (x0 ))
= lim sup
.
j→∞
2εj
2ε
ε→0
(4.18)
lim
Items (iii) and (iv) in Lemma 3.3 and the Coarea formula yield for a subsequence not relabeled for convenience that for L1 -a.e. s ∈ (0, 1)
ˆ
1
1 + |u − wεj | dH1 = 0,
(4.19)
lim
j εj ∂B (x )∩{u6=w }
εj
sεj
0
µ ∂Bsεj (x0 ) = H1 ∂Bsεj (x0 ) ∩ Jwεj = 0.
(4.20)
We choose zj ∈ SBDp (Bsεj (x0 )) such that zj = u around ∂Bsεj (x0 ) and
F (zj , Bsεj (x0 )) ≤ m(u, Bsεj (x0 )) + ε2j ,
and define
(
zj
ζj :=
wε j
Bsεj (x0 )
Bεj (x0 ) \ Bsεj (x0 ).
The definition of zj , the growth conditions in (1.1), and the locality of F
yield
m(wεj , Bεj (x0 )) ≤ F (ζj , Bεj (x0 ))
ˆ
≤ F (zj , Bsεj (x0 )) + β
(1 + |e(wεj )|p ) dx
Bεj (x0 )\Bsεj (x0 )
|
{z
}
(1)
ˆ
ˆ
=:Ij
1
(1 + |[wεj ]|)dH1
(1 + |u − wεj |)dH + β
+β
∂Bsεj (x0 )∩{u6=wεj }
|
(Bεj (x0 )\Bsεj (x0 ))∩Jwεj
{z
}
(2)
=:Ij
|
{z
}
(3)
=:Ij
(1)
(2)
(3)
≤ m(u, Bsεj (x0 )) + ε2j + Ij + Ij + Ij .
(1)
(2)
We note that Ij and Ij are o(εj ) as j → ∞ thanks to Lemma 3.3 (ii) and
(4.19), respectively. Instead, employing Lemma 3.3 (vi) and (4.13) to bound
int_repr10.tex
31
[September 28, 2015]
(3)
Ij
we infer that
(3)
Ij
β
lim sup
≤ lim sup
j→∞ 2εj
j→∞ 2εj
ˆ
(1 + |[u]|)dH1
(Bεj (x0 )\Bsεj (x0 )∩Ju
µ Bεj (x0 ) \ Bsεj (x0 ) ∩ Ju
= (1 − s)β(1 + |[u](x0 )|). (4.21)
= β lim sup
2εj
j→∞
Therefore, by (4.15) we conclude
m(u, Bsεj (x0 ))
m(wεj , Bεj (x0 ))
≤ lim inf
+ (1 − s)β(1 + |[u](x0 )|)
j→∞
j→∞
2εj
2εj
dF (u, ·)
=s 1
(x0 ) + (1 − s)β(1 + |[u](x0 )|).
dH Ju
lim
Estimate (4.16) follows at once by (4.18) and by letting s ↑ 1 in the last
inequality.
Let now (εj )j be a sequence such that
m(wεj , Bεj (x0 ))
m(wε , Bε (x0 ))
= lim inf
.
ε→0
j→∞
2εj
2ε
lim
(4.22)
Let λ ∈ (1, 2), arguing as for (4.19) and (4.20), up to a subsequence depending
on λ and not relabeled for convenience we may assume that for L1 -a.e. s ∈
(0, 1)
ˆ
1
(4.23)
1 + |u − wεj | dH1 = 0,
lim
j→∞ εj ∂B
sλεj (x0 )∩{u6=wεj }
and
µ ∂Bsλεj (x0 ) = H1 ∂Bsλεj (x0 ) ∩ Jwεj = 0.
(4.24)
Given zj ∈ SBDp (Bsλεj (x0 )) with zj = wεj around ∂Bsλεj (x0 ) and such that
F (zj , Bsλεj (x0 )) ≤ m(wεj , Bsλεj (x0 )) + ε2j ,
define
(
zj
ζj :=
u
Bsλεj (x0 )
Bλεj (x0 ) \ Bsλεj (x0 ).
Using ζj as a test field for m(u, Bλεj (x0 )), by the locality of F and its growth
int_repr10.tex
32
[September 28, 2015]
conditions in (1.1)
m(u, Bλεj (x0 )) ≤ F (ζj , Bλεj (x0 )) ≤ m(wεj , Bsλεj (x0 )) + ε2j
ˆ
ˆ
p
(1 + |e(u)| ) dx + β
+β
(1 + |u − wεj |)dH1
∂Bsλεj (x0 )∩{u6=wεj }
Bλεj (x0 )
|
{z
}
(4)
Ij
|
{z
}
(5)
Ij
ˆ
(1 + |[u]|)dH1 .
+β
(Bλεj (x0 )\Bsλεj (x0 ))∩Ju
|
{z
}
(6)
Ij
(4)
(5)
The terms Ij and Ij are o(εj ) by (4.14) and (4.23), respectively. The term
(6)
Ij can be estimated thanks to (4.13). Hence, we get by (4.15)
m(u, Bλεj (x0 ))
m(wεj , Bsλεj (x0 ))
dF (u, ·)
(x0 ) = lim sup
≤ lim sup
. (4.25)
1
dH Ju
2λεj
2λεj
j→∞
j→∞
Next, by choosing s ∈ (0, 1) for which (4.23) and (4.24) hold and sλ > 1, we
may use Lemma 4.2(ii) to infer
ˆ
(1 + |e(wεj )|p )dx
m(wεj , Bsλεj (x0 )) ≤m(wεj , Bεj (x0 )) + β
Bsλεj (x0 )\Bεj (x0 )
ˆ
(1 + |[wεj ]|)dH1 .
+β
(4.26)
(Bsλεj (x0 )\Bεj (x0 ))∩Jwεj
Clearly, the first integral is o(εj ) by Lemma 3.3 (ii), while the other one can
(3)
be dealt with as Ij in (4.21). Thus, (4.25) and (4.26) give
m(wεj , Bεj (x0 ))
dF (u, ·)
1
(x0 ) ≤ lim
+ (sλ − 1)β(1 + |[u](x0 )|).
1
dH Ju
λ j→∞
2εj
In conclusion, by taking into account (4.22), we deduce (4.17) by taking first
the limit as s ↑ 1, for s ∈ (0, 1) chosen as explained above, and then as λ ↓ 1
in the latter inequality.
We are now ready to show that the function g in (4.2) is the surface energy
density of F . This task shall be accomplished by proving two inequalities.
Lemma 4.9. For H1 -a.e. x0 ∈ Ju ,
dF (u, ·)
(x0 ) ≤ g(x0 , u+ (x0 ), u− (x0 ), νu (x0 ))
dH1 Ju
where g was defined in (4.2).
int_repr10.tex
33
[September 28, 2015]
Proof. We consider the same x0 as in Lemma 4.8. In view of (4.12) and the
definition of g in (4.2) it suffices to show that
m(wε , Bε (x0 ))
m(ux0 , Bε (x0 ))
≤ lim sup
,
ε→0
2ε
2ε
ε→0
lim
(4.27)
where wε is the function introduced in Lemma 4.8. To prove such a claim
consider any sequence (εj )j , we have that for L1 -a.e. s ∈ (0, 1)
(4.28)
µ ∂Bsεj (x0 ) = H1 ∂Bsεj (x0 ) ∩ Jwj = 0,
where we have set wj := wεj .
Fix s ∈ (0, 1) as above and a test field zj ∈ SBDp (Bsεj (x0 )) with zj = ux0
on ∂Bsεj (x0 ) such that
F (zj , Bsεj (x0 )) ≤ m(ux0 , Bsεj (x0 )) + ε2j .
Consider a cut-off function ϕ ∈ Cc∞ (Bεj (x0 ), [0, 1]) such that ϕ ≡ 1 on
2
Bsεj (x0 ) and k∇ϕkL∞ ≤ (1−s)ε
. Define ζj := ϕ zj + (1 − ϕ)wj , with the
j
convention that zj is extended equal to ux0 outside Bsεj (x0 ). Therefore, by
using ζj as a test field for m(wj , Bεj (x0 )) we infer from the growth condition
in (1.1) and the locality of F
m(wj , Bεj (x0 )) ≤ F (ζj , Bεj (x0 )) ≤ F (zj , Bsεj (x0 ))
ˆ
ˆ
C
p
|wj − ux0 |p dx
(1 + |e(wj )| ) dx +
+C
p
((1
−
s)ε
)
j
Bεj (x0 )\Bsεj (x0 )
Bεj (x0 )\Bsεj (x0 )
{z
} |
{z
}
|
(7)
(8)
=:Ij
=:Ij
ˆ
+ C H1 (Bεj (x0 ) \ Bsεj (x0 )) ∩ Jζj + C
{z
}
|
(9)
|
=:Ij
|[ζj ]|dH1
(Bεj (x0 )\Bsεj (x0 ))∩Jζj
{z
}
(10)
=:Ij
(7)
(8)
(9)
(10)
≤ m(ux0 , Bsεj (x0 )) + ε2j + Ij + Ij + Ij + Ij
, (4.29)
with C = C(β, p) > 0.
(7)
(8)
By taking into account Lemma 3.3 (ii) and (v) we deduce that Ij +Ij =
o(εj ) as j → ∞. Moreover, as
H1 ((Bεj (x0 ) \ Bsεj (x0 )) ∩ Jζj \ (Jux0 ∪ Jwj )) = 0,
item (i) in Lemma 3.3 together with (4.13) give
(9)
Ij
lim sup
≤ C(1 − s)(1 + |[u](x0 )|).
j→∞ 2εj
int_repr10.tex
34
[September 28, 2015]
Furthermore, for H1 -a.e. x ∈ Jζj ∩ (Bεj (x0 ) \ Bsεj (x0 )) it holds
|[ζj ]| ≤ |[ux0 ]|χJux0 ∩Jζj + |[wj ]|χJwj ∩Jζj ≤ 2|[ux0 ]|χJζj + |[wj ] − [ux0 ]|χJwj .
In turn the latter inequality implies by (4.13) and (4.28)
(10)
Ij
lim sup
≤ C(1 − s)|[u](x0 )|
2εj
j→∞
ˆ
1
(|[wj ] − [u](x0 )|) dH1
+ C lim sup
j→∞ 2εj (Bε (x0 )\Bsε (x0 ))∩Jζ
j
j
j
≤ C(1 − s)|[u](x0 )|,
thanks to item (vi) in Lemma 3.3.
Finally, we obtain from (4.29)
lim inf
j→∞
m(wj , Bεj (x0 ))
m(ux0 , Bsεj (x0 ))
≤ s lim sup
+C(1−s)(1+|[u](x0 )|)
2εj
2sεj
j→∞
m(ux0 , Bε (x0 ))
≤ s lim sup
+ C(1 − s)(1 + |[u](x0 )|),
2ε
ε→0
and the claim in (4.27) follows at once by letting s → 1 in the inequality
above.
The reverse inequality is established arguing in an analogous fashion,
therefore we provide a more concise proof.
Lemma 4.10. For H1 -a.e. x0 ∈ Ju ,
dF (u, ·)
(x0 ) ≥ g(x0 , u+ (x0 ), u− (x0 ), νu (x0 ))
1
dH Ju
where g was defined in (4.2).
Proof. We consider the same points x0 as in Lemma 4.8. Take any infinitesimal sequence (εj )j such that
m(ux0 , Bεj (x0 ))
,
j→∞
2εj
g(x0 , u+ (x0 ), u− (x0 ), νu (x0 )) = lim
and recall that (4.28) is valid for L1 -a.e. s ∈ (0, 1) (as usual wj = wεj ).
Having fixed such an s, let zj ∈ SBDp (Bs εj (x0 )) with zj = wj on ∂Bsεj (x0 )
be such that
F (zj , Bsεj (x0 )) ≤ m(wj , Bsεj (x0 )) + ε2j .
int_repr10.tex
35
[September 28, 2015]
Let ϕ ∈ Cc∞ (Bεj (x0 ), [0, 1]) be a cut-off function such that ϕ ≡ 1 on Bsεj (x0 )
2
and k∇ϕkL∞ ≤ (1−s)ε
. Define ζj := ϕ zj + (1 − ϕ)ux0 , with the convention
j
that zj is extended equal to wj outside Bsεj (x0 ). By using ζj as a test field for
m(ux0 , Bεj (x0 )) we infer from the growth condition in (1.1) and the locality
of F
m(ux0 , Bεj (x0 )) ≤ F (ζj , Bεj (x0 )) ≤ m(wj , Bsεj (x0 )) + ε2j
ˆ
ˆ
C
p
(1+|e(wj )| ) dx+
|wj −ux0 |p dx
+C
p
((1
−
s)ε
)
j
Bεj (x0 )\Bs εj (x0 )
Bεj (x0 )\Bsεj (x0 )
ˆ
(1 + |[ζj ]|)dH1 ,
+C
(Bεj (x0 )\Bsεj (x0 ))∩Jζj
where C = C(β, p) > 0. Arguing as in the corresponding estimate in
Lemma 4.9 (cf. (4.29)), and by taking into account the choice of (εj )j we
conclude that
m(wj , Bsεj (x0 ))
+C(1−s)(1+|[u](x0 )|)
j→∞
2εj
dF (u, ·)
=s 1
(x0 ) + C(1 − s)(1 + |[u](x0 )|).
dH Ju
g(x0 , u+ (x0 ), u− (x0 ), νu (x0 )) ≤ lim inf
The last equality follows from (4.12). The conclusion is achieved by letting
s ↑ 1 in the last inequality, with s ∈ (0, 1) satisfying (4.28).
4.4
Proof of Theorem 1.1
Proof of Theorem 1.1. The conclusion straightforwardly follows by Lemmata
4.6, 4.7, 4.9, and 4.10.
Proposition 4.11. The assertion in Theorem 1.1 holds also if property (iv)
is replaced by the weaker
(iv’) There are α, β > 0 such that for any u ∈ SBDp (Ω), any B ∈ B(Ω),
ˆ
α
|e(u)|p dx + H1 (Ju ∩ B) ≤ F (u, B)
B
ˆ
ˆ
p
1
≤β
(|e(u)| + 1)dx +
(1 + |[u]|)dH .
Ju ∩B
B
Proof. Given F satisfying properties (i)-(iii) and (iv’), we define for δ > 0 a
functional Fδ : SBDp (Ω) × B(Ω) → [0, ∞) by
ˆ
Fδ (u, B) := F (u, B) + δ
|[u]|dH1 ,
Ju ∩B
int_repr10.tex
36
[September 28, 2015]
for u ∈ SBDp (Ω) and B ∈ B(Ω). Since Fδ satisfies properties (i)-(iv), there
are two functions f and gδ such that Fδ can be represented as in (1.2). The
family of functionals Fδ is pointwise increasing in δ, therefore there exists the
pointwise limit g of gδ as δ → 0. We conclude that the representation (1.2)
holds for F with densities f and g.
Remark 4.12. Since F is lower semicontinuous on W 1,p , the integrand f is
quasiconvex [1, 36]. Since F is lower semicontinuous on piecewise constant
functions, g is BV -elliptic [2, 3].
Remark 4.13. If the functional F additionally obeys
F (u + I, B) = F (I, B),
for every u ∈ SBDp (Ω), every ball B ⊂ Ω, and every affine function I such
that e(I) = 0, then there are two functions f : Ω × R2×2 → [0, ∞) and
g : Ω × R2 × S 1 → [0, ∞) such that
ˆ
ˆ
g(x, [u](x), νu (x))dH1 .
f (x, e(u(x)))dx +
F (u, B) =
B∩Ju
B
Remark 4.14. A growth condition on the volume part of the type of (1.1)
alone does not force the energy density to depend only on e(u). As an example, the integrand f : R2×2 → [0, ∞) defined by
q
2
2
2 2
+ ξ21
) + 1 − 2 det(ξ)
f (ξ) := (ξ11 + ξ22 ) + (ξ12
satisfies
1
1
|ξ + ξ T |2 ≤ f (ξ) ≤ |ξ + ξ T |2 + 1
8
4
2×2
for every ξ ∈ R , but evidently f (ξ) depends also on the skew-symmetric
part ξ − ξ T . At the same time, f is quasiconvex. We do not know if there is
g such that the functional F defined as in (1.2) satisfies the growth condition
(1.1) and is lower semicontinuous.
Acknowledgments
F. Iurlano wishes to thank Gianni Dal Maso for an interesting discussion.
This work was partially supported by the Deutsche Forschungsgemeinschaft
through the Sonderforschungsbereich 1060 “The mathematics of emergent
effects”, project A6. S. Conti thanks the University of Florence for the warm
hospitality of the DiMaI “Ulisse Dini”, where part of this work was carried
out. M. Focardi and F. Iurlano are members of the Gruppo Nazionale per
l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the
Istituto Nazionale di Alta Matematica (INdAM).
int_repr10.tex
37
[September 28, 2015]
References
[1] E. Acerbi and N. Fusco, Semicontinuity problems in the calculus of
variations, Arch. Rat. Mech. Anal., 86 (1984), pp. 125–145.
[2] L. Ambrosio and A. Braides, Functionals defined on partitions in
sets of finite perimeter. I. Integral representation and Γ-convergence, J.
Math. Pures Appl. (9), 69 (1990), pp. 285–305.
[3]
, Functionals defined on partitions in sets of finite perimeter. II.
Semicontinuity, relaxation and homogenization, J. Math. Pures Appl.
(9), 69 (1990), pp. 307–333.
[4] L. Ambrosio, A. Coscia, and G. Dal Maso, Fine properties of
functions with bounded deformation, Arch. Rational Mech. Anal., 139
(1997), pp. 201–238.
[5] L. Ambrosio, N. Fusco, and D. Pallara, Functions of bounded
variation and free discontinuity problems, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 2000.
[6] G. Anzellotti and M. Giaquinta, Existence of the displacement
field for an elastoplastic body subject to Hencky’s law and von Mises
yield condition, Manuscripta Math., 32 (1980), pp. 101–136.
[7] G. Bellettini, A. Coscia, and G. Dal Maso, Compactness and
lower semicontinuity properties in SBD(Ω), Math. Z., 228 (1998),
pp. 337–351.
[8] G. Bouchitté and G. Dal Maso, Integral representation and relaxation of convex local functionals on BV(Ω), Ann. Scuola Norm. Sup.
Pisa Cl. Sci. (4), 20 (1993), pp. 483–533.
[9] G. Bouchitté, I. Fonseca, G. Leoni, and L. Mascarenhas, A
global method for relaxation in W 1,p and in SBV p , Arch. Ration. Mech.
Anal., 165 (2002), pp. 187–242.
[10] G. Bouchitté, I. Fonseca, and L. Mascarenhas, A global method
for relaxation, Arch. Rational Mech. Anal., 145 (1998), pp. 51–98.
[11] A. Braides and V. Chiadò Piat, Integral representation results for
functionals defined on SBV(Ω; Rm ), J. Math. Pures Appl. (9), 75 (1996),
pp. 595–626.
int_repr10.tex
38
[September 28, 2015]
[12] G. Buttazzo and G. Dal Maso, Γ-limits of integral functionals, J.
Analyse Math., 37 (1980), pp. 145–185.
[13]
, A characterization of nonlinear functionals on Sobolev spaces
which admit an integral representation with a Carathéodory integrand,
J. Math. Pures Appl. (9), 64 (1985), pp. 337–361.
[14]
, Integral representation and relaxation of local functionals, Nonlinear Anal., 9 (1985), pp. 515–532.
[15] L. Carbone and C. Sbordone, Some properties of Γ-limits of integral functionals, Ann. Mat. Pura Appl. (4), 122 (1979), pp. 1–60.
[16] A. Chambolle, An approximation result for special functions with
bounded deformation, J. Math. Pures Appl. (9), 83 (2004), pp. 929–954.
[17]
, Addendum to: “An approximation result for special functions with
bounded deformation” [16], J. Math. Pures Appl. (9), 84 (2005), pp. 137–
145.
[18] A. Chambolle, S. Conti, and G. Francfort, Korn-Poincaré inequalities for functions with a small jump set, Preprint hal-01091710v1,
(2014).
[19] A. Chambolle, A. Giacomini, and M. Ponsiglione, Piecewise
rigidity, J. Funct. Anal., 244 (2007), pp. 134–153.
[20] S. Conti, M. Focardi, and F. Iurlano, Existence theorem for the
Griffith fracture model in dimension two, In preparation.
[21] S. Conti, M. Focardi, and F. Iurlano, Which special functions of bounded deformation have bounded variation?, Preprint
arXiv:1502.07464, (2015).
[22] S. Conti and B. Schweizer, Rigidity and gamma convergence for
solid-solid phase transitions with SO(2) invariance, Comm. Pure Appl.
Math., 59 (2006), pp. 830–868.
[23]
, A sharp-interface limit for a two-well problem in geometrically
linear elasticity, Arch. Ration. Mech. Anal., 179 (2006), pp. 413–452.
[24] G. Dal Maso, Integral representation on BV(Ω) of Γ-limits of variational integrals, Manuscripta Math., 30 (1979/80), pp. 387–416.
int_repr10.tex
39
[September 28, 2015]
[25]
, An introduction to Γ-convergence, Progress in Nonlinear Differential Equations and their Applications, 8, Birkhäuser Boston Inc., Boston,
MA, 1993.
[26]
, Generalised functions of bounded deformation, J. Eur. Math. Soc.
(JEMS), 15 (2013), pp. 1943–1997.
[27] G. Dal Maso and L. Modica, A general theory of variational functionals, in Topics in functional analysis, 1980–81, Quaderni, Scuola
Norm. Sup. Pisa, Pisa, 1981, pp. 149–221.
[28]
, Integral functionals determined by their minima, Rend. Sem. Mat.
Univ. Padova, 76 (1986), pp. 255–267.
[29] E. De Giorgi, Sulla convergenza di alcune successioni d’integrali del
tipo dell’area, Rend. Mat. (6), 8 (1975), pp. 277–294. Collection of articles dedicated to Mauro Picone on the occasion of his ninetieth birthday.
[30] E. De Giorgi, M. Carriero, and A. Leaci, Existence theorem for
a minimum problem with free discontinuity set, Arch. Rational Mech.
Anal., 108 (1989), pp. 195–218.
[31] F. Ebobisse and R. Toader, A note on the integral representation
representation of functionals in the space SBD(Ω), Rend. Mat. Appl.
(7), 23 (2003), pp. 189–201 (2004).
[32] M. Friedrich, A Korn-Poincaré-type inequality for special functions
of bounded deformation, preprint, (2015).
[33]
, A Korn-type inequality in SBD for functions with small jump sets,
preprint, (2015).
[34] G. Friesecke, R. D. James, and S. Müller, A theorem on geometric rigidity and the derivation of nonlinear plate theory from threedimensional elasticity, Comm. Pure Appl. Math., 55 (2002), pp. 1461–
1506.
[35] R. Kohn and R. Temam, Dual spaces of stresses and strains, with
applications to Hencky plasticity, Appl. Math. Optim., 10 (1983), pp. 1–
35.
[36] P. Marcellini, Approximation of quasiconvex functions, and lower
semicontinuity of multiple integrals, Manuscripta Math., 51 (1985),
pp. 1–28.
int_repr10.tex
40
[September 28, 2015]
[37] J. A. Nitsche, On Korn’s second inequality, RAIRO Anal. Numér., 15
(1981), pp. 237–248.
[38] C. Sbordone, Su alcune applicazioni di un tipo di convergenza variazionale, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 2 (1975), pp. 617–
638.
[39] P.-M. Suquet, Sur un nouveau cadre fonctionnel pour les équations
de la plasticité, C. R. Acad. Sci. Paris Sér. A-B, 286 (1978), pp. A1129–
A1132.
[40] R. Temam, Problèmes mathématiques en plasticité, vol. 12 of Méthodes
Mathématiques de l’Informatique, Gauthier-Villars, Montrouge, 1983.
[41] R. Temam and G. Strang, Functions of bounded deformation, Arch.
Rational Mech. Anal., 75 (1980/81), pp. 7–21.
int_repr10.tex
41
[September 28, 2015]